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Basic Business Statistics
10th Edition
Chapter 10
Two-Sample Tests
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-1
Learning Objectives
In this chapter, you learn:
 How to use hypothesis testing for comparing the
difference between
 The means of two independent populations
 The means of two related populations
 The proportions of two independent populations
 The variances of two independent populations
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-2
Two-Sample Tests
Two-Sample Tests
Population
Means,
Independent
Samples
Means,
Related
Samples
Population
Proportions
Population
Variances
Examples:
Population 1 vs.
independent
Population 2
Same population
before vs. after
treatment
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Proportion 1 vs.
Proportion 2
Variance 1 vs.
Variance 2
Chap 10-3
Difference Between Two Means
Population means,
independent
samples
*
σ1 and σ2 known
σ1 and σ2 unknown,
assumed equal
σ1 and σ2 unknown,
not assumed equal
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Goal: Test hypothesis or form
a confidence interval for the
difference between two
population means, μ1 – μ2
The point estimate for the
difference is
X1 – X2
Chap 10-4
Independent Samples
Population means,
independent
samples
*
σ1 and σ2 known
σ1 and σ2 unknown,
assumed equal
σ1 and σ2 unknown,
not assumed equal
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
 Different data sources
 Unrelated
 Independent
 Sample selected from one
population has no effect on the
sample selected from the other
population
 Use the difference between 2
sample means
 Use Z test, a pooled-variance t
test, or a separate-variance t
test
Chap 10-5
Difference Between Two Means
Population means,
independent
samples
*
σ1 and σ2 known
Use a Z test statistic
σ1 and σ2 unknown,
assumed equal
Use Sp to estimate unknown
σ , use a t test statistic and
pooled standard deviation
σ1 and σ2 unknown,
not assumed equal
Use S1 and S2 to estimate
unknown σ1 and σ2, use a
separate-variance t test
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-6
σ1 and σ2 Known
Population means,
independent
samples
σ1 and σ2 known
σ1 and σ2 unknown,
assumed equal
σ1 and σ2 unknown,
not assumed equal
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Assumptions:
*
 Samples are randomly and
independently drawn
 Population distributions are
normal or both sample sizes
are
30
 Population standard
deviations are known
Chap 10-7
σ1 and σ2 Known
(continued)
Population means,
independent
samples
σ1 and σ2 known
σ1 and σ2 unknown,
assumed equal
σ1 and σ2 unknown,
not assumed equal
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
When σ1 and σ2 are known and
both populations are normal or
both sample sizes are at least 30,
the test statistic is a Z-value…
*
…and the standard error of
X1 – X2 is
σ X1  X2 
2
1
2
σ
σ2

n1
n2
Chap 10-8
σ1 and σ2 Known
(continued)
Population means,
independent
samples
σ1 and σ2 known
σ1 and σ2 unknown,
assumed equal
The test statistic for
μ1 – μ2 is:
*

X
Z
1

 X 2   μ1  μ2 
2
1
2
σ
σ2

n1
n2
σ1 and σ2 unknown,
not assumed equal
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-9
Hypothesis Tests for
Two Population Means
Two Population Means, Independent Samples
Lower-tail test:
Upper-tail test:
Two-tail test:
H0: μ1
μ2
H1: μ1 < μ2
H0: μ1 ≤ μ2
H1: μ1 > μ2
H0: μ1 = μ2
H1: μ1 ≠ μ2
i.e.,
i.e.,
i.e.,
H0: μ1 – μ2
0
H1: μ1 – μ2 < 0
H0: μ1 – μ2 ≤ 0
H1: μ1 – μ2 > 0
H0: μ1 – μ2 = 0
H1: μ1 – μ2 ≠ 0
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-10
Hypothesis tests for μ1 – μ2
Two Population Means, Independent Samples
Lower-tail test:
Upper-tail test:
Two-tail test:
H0: μ1 – μ2
0
H1: μ1 – μ2 < 0
H0: μ1 – μ2 ≤ 0
H1: μ1 – μ2 > 0
H0: μ1 – μ2 = 0
H1: μ1 – μ2 ≠ 0
a
a
-za
Reject H0 if Z < -Za
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
za
Reject H0 if Z > Za
a/2
-za/2
a/2
za/2
Reject H0 if Z < -Za/2
or Z > Za/2
Chap 10-11
Confidence Interval,
σ1 and σ2 Known
Population means,
independent
samples
σ1 and σ2 known
σ1 and σ2 unknown,
assumed equal
The confidence interval for
μ1 – μ2 is:
*


2
1
2
σ
σ2
X1  X 2  Z

n1
n2
σ1 and σ2 unknown,
not assumed equal
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-12
σ1 and σ2 Unknown,
Assumed Equal
Assumptions:
Population means,
independent
samples
 Samples are randomly and
independently drawn
σ1 and σ2 known
σ1 and σ2 unknown,
assumed equal
σ1 and σ2 unknown,
not assumed equal
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
*
 Populations are normally
distributed or both sample
sizes are at least 30
 Population variances are
unknown but assumed equal
Chap 10-13
σ1 and σ2 Unknown,
Assumed Equal
(continued)
Forming interval estimates:
Population means,
independent
samples
σ1 and σ2 known
σ1 and σ2 unknown,
assumed equal
σ1 and σ2 unknown,
not assumed equal
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
*
 The population variances
are assumed equal, so use
the two sample variances
and pool them to
estimate the common σ2
 the test statistic is a t value
with (n1 + n2 – 2) degrees
of freedom
Chap 10-14
σ1 and σ2 Unknown,
Assumed Equal
(continued)
Population means,
independent
samples
The pooled variance is
σ1 and σ2 known
σ1 and σ2 unknown,
assumed equal
*
S
2
p

n1  1S

 n2  1S2
(n1  1)  (n2  1)
2
1
2
σ1 and σ2 unknown,
not assumed equal
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-15
σ1 and σ2 Unknown,
Assumed Equal
(continued)
The test statistic for
μ1 – μ2 is:
Population means,
independent
samples

X  X   μ  μ 
t
1
σ1 and σ2 known
σ1 and σ2 unknown,
assumed equal
σ1 and σ2 unknown,
not assumed equal
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
2
1
2
1 1 
S   
 n1 n2 
2
p
*
Where t has (n1 + n2 – 2) d.f.,
and
S
2
p
2
2

n1  1S1  n2  1S2

(n1  1)  (n2  1)
Chap 10-16
Confidence Interval,
σ1 and σ2 Unknown
Population means,
independent
samples
The confidence interval for
μ1 – μ2 is:
X  X   t
σ1 and σ2 known
σ1 and σ2 unknown,
assumed equal
σ1 and σ2 unknown,
not assumed equal
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
1
2
*
n1 n2 -2
1 1 
S   
 n1 n2 
2
p
Where
2
2




n

1
S

n

1
S
1
2
2
S2  1
p
(n1  1)  (n2  1)
Chap 10-17
Pooled-Variance t Test: Example
You are a financial analyst for a brokerage firm. Is there a
difference in dividend yield between stocks listed on the
NYSE & NASDAQ? You collect the following data:
NYSE NASDAQ
Number
21
25
Sample mean
3.27
2.53
Sample std dev 1.30
1.16
Assuming both populations are
approximately normal with
equal variances, is
there a difference in average
yield ( = 0.05)?
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-18
Calculating the Test Statistic
The test statistic is:

X  X   μ  μ 
t

1
2
1
1 1
S   
 n1 n2 
2
p
2
3.27  2.53   0
1 
 1
1.5021 

 21 25 
2
2
2
2








n

1
S

n

1
S
21

1
1.30

25

1
1.16
1
2
2
S2  1

p
(n1  1)  (n2  1)
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
(21 - 1)  (25  1)
 2.040
 1.5021
Chap 10-19
Solution
H0: μ1 - μ2 = 0 i.e. (μ1 = μ2)
H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2)
 = 0.05
df = 21 + 25 - 2 = 44
Critical Values: t = ± 2.0154
Reject H0
.025
-2.0154
Reject H0
.025
0 2.0154
t
2.040
Test Statistic:
Decision:
3.27  2.53
t
 2.040 Reject H0 at a = 0.05
1 
 1
Conclusion:
1.5021  

 21 25 
There is evidence of a
difference in means.
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-20
σ1 and σ2 Unknown,
Not Assumed Equal
Assumptions:
Population means,
independent
samples
 Samples are randomly and
independently drawn
 Populations are normally
distributed or both sample
sizes are at least 30
σ1 and σ2 known
σ1 and σ2 unknown,
assumed equal
σ1 and σ2 unknown,
not assumed equal
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
*
 Population variances are
unknown but cannot be
assumed to be equal
Chap 10-21
σ1 and σ2 Unknown,
Not Assumed Equal
(continued)
Population means,
independent
samples
Forming the test statistic:
 The population variances
are not assumed equal, so
include the two sample
variances in the computation
of the t-test statistic
σ1 and σ2 known
σ1 and σ2 unknown,
assumed equal
σ1 and σ2 unknown,
not assumed equal
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
*
 the test statistic is a t value
with v degrees of freedom
(see next slide)
Chap 10-22
σ1 and σ2 Unknown,
Not Assumed Equal
(continued)
Population means,
independent
samples
The number of degrees of
freedom is the integer
portion of:
σ1 and σ2 known
σ1 and σ2 unknown,
assumed equal
σ1 and σ2 unknown,
not assumed equal
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
*
2
S
S2 



 n

n
2 
   12
2
2 2
 S1   S 2 

 

 n   n 
 1   2 
n1  1
n2  1
2
1
2
Chap 10-23
σ1 and σ2 Unknown,
Not Assumed Equal
(continued)
Population means,
independent
samples
The test statistic for
μ1 – μ2 is:

X  X   μ  μ 
t
σ1 and σ2 known
1
σ1 and σ2 unknown,
assumed equal
σ1 and σ2 unknown,
not assumed equal
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
2
2
1
*
1
2
2
2
S S

n1 n2
Chap 10-24
Related Populations
Tests Means of 2 Related Populations
Related
samples



Paired or matched samples
Repeated measures (before/after)
Use difference between paired values:
Di = X1i - X2i
 Eliminates Variation Among Subjects
 Assumptions:
 Both Populations Are Normally Distributed
 Or, if not Normal, use large samples
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-25
Mean Difference, σD Known
The ith paired difference is Di , where
Related
samples
Di = X1i - X2i
n
The point estimate for
the population mean
paired difference is D :
D
D
i 1
i
n
Suppose the population
standard deviation of the
difference scores, σD, is known
n is the number of pairs in the paired sample
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-26
Mean Difference, σD Known
(continued)
Paired
samples
The test statistic for the mean
difference is a Z value:
D  μD
Z
σD
n
Where
μD = hypothesized mean difference
σD = population standard dev. of differences
n = the sample size (number of pairs)
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-27
Confidence Interval, σD Known
Paired
samples
The confidence interval for μD is
σD
DZ
n
Where
n = the sample size
(number of pairs in the paired sample)
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-28
Mean Difference, σD Unknown
Related
samples
If σD is unknown, we can estimate the
unknown population standard deviation
with a sample standard deviation:
The sample standard
deviation is
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
n
SD 
2
(D

D
)
 i
i1
n 1
Chap 10-29
Mean Difference, σD Unknown
(continued)
Paired
samples
 Use a paired t test, the test statistic for
D is now a t statistic, with n-1 d.f.:
D  μD
t
SD
n
n
Where t has n - 1 d.f.
and SD is:
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
SD 
2
(D

D
)
 i
i1
n 1
Chap 10-30
Confidence Interval, σD Unknown
Paired
samples
The confidence interval for μD is
SD
D  t n1
n
n
where
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
SD 
 (D  D)
i1
2
i
n 1
Chap 10-31
Hypothesis Testing for
Mean Difference, σD Unknown
Paired Samples
Lower-tail test:
Upper-tail test:
Two-tail test:
H0: μD
0
H1: μD < 0
H0: μD ≤ 0
H1: μD > 0
H0: μD = 0
H1: μD ≠ 0
a
a
-ta
ta
Reject H0 if t < -ta
Reject H0 if t > ta
Where t has n - 1 d.f.
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
a/2
-ta/2
a/2
ta/2
Reject H0 if t < -ta/2
or t > ta/2
Chap 10-32
Paired t Test Example
 Assume you send your salespeople to a “customer
service” training workshop. Has the training made a
difference in the number of complaints? You collect
the following data:
Number of Complaints:
(2) - (1)
Salesperson Before (1) After (2)
Difference, Di
C.B.
T.F.
M.H.
R.K.
M.O.
6
20
3
0
4
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
4
6
2
0
0
- 2
-14
- 1
0
- 4
-21
D =
Di
n
= -4.2
SD 
2
(D

D
)
 i
n 1
 5.67
Chap 10-33
Paired t Test: Solution
 Has the training made a difference in the number of
complaints (at the 0.01 level)?
H0: μD = 0
H1: μD
0
=
D = - 4.2
.01
Critical
Value = ± 4.604
d.f. = n - 1 = 4
Reject
Reject
/2
/2
- 4.604
4.604
- 1.66
Decision: Do not reject H0
(t stat is not in the reject region)
Test Statistic:
D  μD  4.2  0
t

 1.66
SD / n 5.67/ 5
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Conclusion: There is not a
significant change in the
number of complaints.
Chap 10-34
Two Population Proportions
Population
proportions
Goal: test a hypothesis or form a
confidence interval for the difference
between two population proportions,
 1 – 2
Assumptions:
n1 1
5 , n1(1- 1)
5
n2 2
5 , n2(1- 2)
5
The point estimate for
the difference is
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
p1  p2
Chap 10-35
Two Population Proportions
Population
proportions
Since we begin by assuming the null
hypothesis is true, we assume 1 = 2
and pool the two sample estimates
The pooled estimate for the
overall proportion is:
X1  X2
p
n1  n2
where X1 and X2 are the numbers from
samples 1 and 2 with the characteristic of
interest
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-36
Two Population Proportions
(continued)
The test statistic for
p1 – p2 is a Z statistic:
Population
proportions
Z
where
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
p
 p1  p2    π1  π2 
1 1
p (1 p)   
 n1 n2 
X1  X2
X
X
, p1  1 , p 2  2
n1  n2
n1
n2
Chap 10-37
Confidence Interval for
Two Population Proportions
Population
proportions
The confidence interval for
1 – 2 is:
 p1  p2 
p1(1 p1 ) p2 (1 p2 )
Z

n1
n2
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-38
Hypothesis Tests for
Two Population Proportions
Population proportions
Lower-tail test:
Upper-tail test:
Two-tail test:
H0: 1
2
H1: 1 < 2
H0: 1 ≤ 2
H1: 1 > 2
H0: 1 = 2
H1: 1 ≠ 2
i.e.,
i.e.,
i.e.,
H0: 1 – 2
0
H1: 1 – 2 < 0
H0: 1 – 2 ≤ 0
H1: 1 – 2 > 0
H0: 1 – 2 = 0
H1: 1 – 2 ≠ 0
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-39
Hypothesis Tests for
Two Population Proportions
(continued)
Population proportions
Lower-tail test:
Upper-tail test:
Two-tail test:
H0: 1 – 2
0
H1: 1 – 2 < 0
H0: 1 – 2 ≤ 0
H1: 1 – 2 > 0
H0: 1 – 2 = 0
H1: 1 – 2 ≠ 0
a
a
-za
Reject H0 if Z < -Za
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
za
Reject H0 if Z > Za
a/2
-za/2
a/2
za/2
Reject H0 if Z < -Za/2
or Z > Za/2
Chap 10-40
Example:
Two population Proportions
Is there a significant difference between the
proportion of men and the proportion of
women who will vote Yes on Proposition A?
 In a random sample, 36 of 72 men and 31 of
50 women indicated they would vote Yes
 Test at the .05 level of significance
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-41
Example:
Two population Proportions
(continued)
 The hypothesis test is:
H0: 1 – 2 = 0 (the two proportions are equal)
H1: 1 – 2 ≠ 0 (there is a significant difference between proportions)
 The sample proportions are:
 Men:
p1 = 36/72 = .50
 Women:
p2 = 31/50 = .62
 The pooled estimate for the overall proportion is:
X1  X 2 36  31 67
p


 .549
n1  n2
72  50 122
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-42
Example:
Two population Proportions
(continued)
The test statistic for 1 – 2 is:
z

 p1  p2     1   2 
1 1
p (1  p)   
 n1 n2 
 .50  .62    0
1 
 1
.549 (1  .549) 


72
50


Reject H0
.025
.025
-1.96
-1.31
  1.31
Critical Values = ±1.96
For
= .05
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Reject H0
1.96
Decision: Do not reject H0
Conclusion: There is not
significant evidence of a
difference in proportions
who will vote yes between
men and women.
Chap 10-43
Hypothesis Test for 2
Ho:  25
df = 15
2
Ha:  25
.05
2
.95
Excel:
=Chiinv(0.95,15)=7.2609
=Chiinv(0.05,15)=24.996
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
.05
0
7.26094
24.9958
Chap 10-44
Hypothesis Test for 2
df = 15
If

2
 7.26094 or

2
 24.9958, reject Ho.
If 7.26094    24.9958, do not reject Ho.
2
.05
.95

2

 n  1 S 2

2

15 281
.
25
 16.86
.05
0
7.26094
24.9958
Since

2
 1686
.   .05,15  24.9958,
2
do not reject Ho.
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-45
Hypothesis Tests for Variances
Tests for Two
Population
Variances
*
F test statistic
H0: σ12 = σ22
H1: σ12 ≠ σ22
Two-tail test
H0: σ12
σ22
H1: σ12 < σ22
Lower-tail test
H0: σ12 ≤ σ22
H1: σ12 > σ22
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Upper-tail test
Chap 10-46
F distribution
 Test for the Difference in 2 Independent
Populations
 Parametric Test Procedure
 Assumptions

Both populations are normally distributed

Test is not robust to this violation
 Samples are randomly and independently drawn
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-47
F Test for Two Population Variances
1
p.372
S 
v1
F

2
S 
v2
2
1
2
2
2
1
2
2
dfnumerator   1  n1  1
dfdeno min ator   1  n2  1
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-48
Hypothesis Tests for Variances
(continued)
Tests for Two
Population
Variances
F test statistic
The F test statistic is:
*
2
1
2
2
S
F
S
S12 = Variance of Sample 1
n1 - 1 = numerator degrees of freedom
S22 = Variance of Sample 2
n2 - 1 = denominator degrees of freedom
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-49
Critical F value for the lower tail (1-)
 F distribution is nonsymmetric. This can be a problem when we
conducting a two-tailed-test and want to determine the critical value for
the lower tail.
 This dilemma can be solved by using Formula
F1 ,v1 ,v2 
F1 ,v1 ,v2 
1
F ,v2 ,v1
2
1
2
2
1
S
 2 2
, (S12  S22 )
S2 / S1 S
1
F ,v2 ,v1
 Which essentially states that the critical F value for the lower tail (1-)
can be solved for by taking the inverse of the F value for the upper tail
().
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-50
The F Distribution
 The F critical value is found from the F table
 There are two appropriate degrees of freedom:
numerator and denominator
S12
F 2
S2
where df1 = n1 – 1 ; df2 = n2 – 1
 In the F table,
 numerator degrees of freedom determine the column
 denominator degrees of freedom determine the row
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-51
Finding the Rejection Region
H0: σ12
σ22
H1: σ12 < σ22
H0: σ12 = σ22
H1: σ12 ≠ σ22
/
0
Reject
H0
F
Do not
reject H0
FL
0
Reject H0 if F < FL
Reject
H0
H0: σ1 ≤ σ2
H1: σ12 > σ22
Do not
reject H0
FU
/
2
2
0
2
Reject H0
2
FL
Do not
reject H0
FU
rejection
region for a
two-tail test is:

F
F
Reject H0
S12
F  2  FU
S2
S12
F  2  FL
S2
Reject H0 if F > FU
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-52
Finding the Rejection Region
(continued)
/
H0: σ12 = σ22
H1: σ12 ≠ σ22
2
/
2
0
Reject
H0
FL
Do not
reject H0
FU
Reject H0
F
To find the critical F values:
1. Find FU from the F table
for n1 – 1 numerator and
n2 – 1 denominator
degrees of freedom
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
1
2. Find FL using the formula: FL 
FU*
Where FU* is from the F table
with n2 – 1 numerator and n1 – 1
denominator degrees of freedom
(i.e., switch the d.f. from FU)
Chap 10-53
F Test: An Example
You are a financial analyst for a brokerage firm. You
want to compare dividend yields between stocks listed
on the NYSE & NASDAQ. You collect the following data:
NYSE
NASDAQ
Number
21
25
Mean
3.27
2.53
Std dev
1.30
1.16
Is there a difference in the
variances between the NYSE
& NASDAQ at the  = 0.05 level?
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-54
F Test: Example Solution
 Form the hypothesis test:
H0: σ21 – σ22 = 0 (there is no difference between variances)
H1: σ21 – σ22 ≠ 0 (there is a difference between variances)
 Find the F critical values for
FU:
 Numerator:
 n1 – 1 = 21 – 1 = 20 d.f.
 Denominator:
 n2 – 1 = 25 – 1 = 24 d.f.
FU = F.025, 20, 24 = 2.33
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
= 0.05:
FL:
 Numerator:
 n2 – 1 = 25 – 1 = 24 d.f.
 Denominator:
 n1 – 1 = 21 – 1 = 20 d.f.
FL = 1/F.025, 24, 20 = 1/2.41
= 0.415
Chap 10-55
F Test: Example Solution
(continued)
 The test statistic is:
H0: σ12 = σ22
H1: σ12 ≠ σ22
S12 1.302
F 2 
 1.256
2
S2 1.16
/2
= .025
/2
= .025
0
Reject H0
 F = 1.256 is not in the rejection
region, so we do not reject H0
Do not
reject H0
FL=0.43
Reject H0
F
FU=2.33
 Conclusion: There is not sufficient evidence
of a difference in variances at
= .05
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-56
Two-Sample Tests in EXCEL
For independent samples:
 Independent sample Z test with variances known:
 Tools | data analysis | z-test: two sample for means
 Pooled variance t test:
 Tools | data analysis | t-test: two sample assuming equal variances
 Separate-variance t test:
 Tools | data analysis | t-test: two sample assuming unequal variances
For paired samples (t test):
 Tools | data analysis | t-test: paired two sample for means
For variances:
 F test for two variances:
 Tools | data analysis | F-test: two sample for variances
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-57
Chapter Summary
 Compared two independent samples
 Performed Z test for the difference in two means
 Performed pooled variance t test for the difference in two
means
 Performed separate-variance t test for difference in two means
 Formed confidence intervals for the difference between two
means
 Compared two related samples (paired samples)
 Performed paired sample Z and t tests for the mean difference
 Formed confidence intervals for the mean difference
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-58
Chapter Summary
(continued)
 Compared two population proportions
 Formed confidence intervals for the difference between two
population proportions
 Performed Z-test for two population proportions
 Performed F tests for the difference between
two population variances
 Used the F table to find F critical values
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 10-59